We present a novel image-based adaptive domain decomposition FEM framework to accelerate the solution of continuum damage mechanics problems. The key idea is to use image-processing techniques in order to identify the moving interface between the healthy subdomain and unhealthy subdomain as damage propagates, and then use an iterative Schur complement approach to efficiently solve the problem. The implementation of the algorithm consists of several modular components. Following the FEM solution of a load increment, the damage detection module is activated, a step that is based on several image-processing operations including colormap manipulation and morphological convolution-based operations. Then, the damage tracking module is invoked, to identify the crack growth direction using geometrical operations and ray casting algorithm. This information is then passed into the domain decomposition module, where the domain is divided into the healthy subdomain which contains only undamaged elements, and the unhealthy subdomain which comprises both damaged and undamaged elements. Continuity between the two regions is restored using penalty constraints. The computational savings of our method stem from the Schur complement, which allows for the iterative solution of the system of equations appertaining only to the unhealthy subdomain. Through an exhaustive comparison between our approach and single domain computations, we demonstrate the accuracy, efficiency, and robustness of the framework. We ensure its compatibility against local and non-local damage laws, structured and unstructured meshes, as well as in cases where different damage paths eventually merge. Since the key novelty lies in using image processing tools to inform the decomposition, our framework can be readily extended beyond damage mechanics and model several classes of non-linear problems such as plasticity and phase-field.

翻译：本文提出了一种新颖的基于图像的自适应域分解有限元框架，用于加速连续损伤力学问题的求解。其核心思想是：利用图像处理技术识别损伤扩展过程中健康子域与非健康子域之间的移动界面，进而采用迭代Schur补方法高效求解问题。该算法的实现包含多个模块化组件：在完成载荷增量的有限元求解后，激活损伤检测模块——该步骤基于包括色彩映射操作和基于形态学卷积运算在内的多种图像处理技术；随后调用损伤追踪模块，通过几何运算与光线投射算法识别裂纹扩展方向。这些信息被传递至域分解模块，将计算域划分为仅包含未损伤单元的健康子域，以及同时包含损伤与未损伤单元的非健康子域，并通过罚约束恢复两区域间的连续性。本方法的计算效率提升源于Schur补技术，该技术允许仅针对非健康子域对应的方程组进行迭代求解。通过将本方法与单域计算进行详尽对比，我们验证了该框架的精确性、高效性与鲁棒性。我们确保了该方法对局部与非局部损伤准则、结构化与非结构化网格的兼容性，以及在多条损伤路径最终合并情况下的适用性。由于本方法的核心创新在于利用图像处理工具指导域分解，该框架可轻松扩展至损伤力学以外的领域，用于模拟塑性、相场等多类非线性问题。